Trust the operating carrier here. When you buy a "K fare from United", this is mapped to another carrier using rules that are embedded in the fare rules. You may want to call LH to confirm that it's an L fare.

(As a striking example of the discrepancy, US Airways has some quite affordable $1200 round trip W fares from the U.S. to Australia, but they seem to book into S/T/L/K on United -- not upgradeable with a GPU.)